Divide the following complex numbers. $ \dfrac{-14+22i}{-2-4i}$
Explanation: We can divide complex numbers by multiplying both numerator and denominator by the denominator's complex conjugate , which is ${-2+4i}$ $ \dfrac{-14+22i}{-2-4i} = \dfrac{-14+22i}{-2-4i} \cdot \dfrac{{-2+4i}}{{-2+4i}} $ We can simplify the denominator using the fact $(a + b) \cdot (a - b) = a^2 - b^2$ $ \dfrac{(-14+22i) \cdot (-2+4i)} {(-2-4i) \cdot (-2+4i)} = \dfrac{(-14+22i) \cdot (-2+4i)} {(-2)^2 - (-4i)^2} $ Evaluate the squares in the denominator and subtract them. $ \dfrac{(-14+22i) \cdot (-2+4i)} {(-2)^2 - (-4i)^2} = $ $ \dfrac{(-14+22i) \cdot (-2+4i)} {4 + 16} = $ $ \dfrac{(-14+22i) \cdot (-2+4i)} {20} $ Note that the denominator now doesn't contain any imaginary unit multiples, so it is a real number, simplifying the problem to complex number multiplication. Now, we can multiply out the two factors in the numerator. $ \dfrac{({-14+22i}) \cdot ({-2+4i})} {20} = $ $ \dfrac{{-14} \cdot {(-2)} + {22} \cdot {(-2) i} + {-14} \cdot {4 i} + {22} \cdot {4 i^2}} {20} $ Evaluate each product of two numbers. $ \dfrac{28 - 44i - 56i + 88 i^2} {20} $ Finally, simplify the fraction. $ \dfrac{28 - 44i - 56i - 88} {20} = \dfrac{-60 - 100i} {20} = -3-5i $